Self-similar solutions with fat tails for Smoluchowski's coagulation equation with locally bounded kernels

TL;DR

This paper proves the existence of self-similar solutions with fat tails for Smoluchowski's coagulation equation with a broad class of homogeneous kernels, extending previous results limited to specific kernels.

Contribution

It establishes the existence of self-similar solutions with fat tails for a wider class of kernels, specifically those that are homogeneous of degree \\gamma \\in [0,1) and satisfy certain bounds.

Findings

Existence of self-similar solutions with decay x^{-(1+\\rho)} for kernels with specified properties.

Extension of known solutions from solvable and diagonal kernels to more general homogeneous kernels.

Construction of continuous weak self-similar profiles with fat tails.

Abstract

The existence of self-similar solutions with fat tails for Smoluchowski's coagulation equation has so far only been established for the solvable and the diagonal kernel. In this paper we prove the existence of such self-similar solutions for continuous kernels $K$ that are homogeneous of degree $\gamma \in [0,1)$ and satisfy $K(x,y) \leq C (x^{\gamma} + y^{\gamma})$. More precisely, for any $\rho \in (\gamma,1)$ we establish the existence of a continuous weak self-similar profile with decay $x^{-(1{+}\rho)}$ as $x \to \infty$.